The **center of mass**, also sometimes called the center of gravity, is typically what we refer to as the geometric position in an object defined by: *the mean position of every section of the object or system, weighted by mass*. In other words, this is a place where the object is **balanced** in our gravitational field.

Below you can see an example of finding the center of mass in the x direction of a system of masses:

Image from LibreTexts

For a system of masses:

Calculus definition:

Another way to format the above formula is with linear mass density:

**Linear mass density** is typically a constant for something that is uniform, so it can be found with an equation like:

However, since AP loves to make us do calculus, we will sometimes see non-uniform objects! This means that the linear mass density would be a function.

Let's try to calculate the center of mass of a uniform rod!

We can begin with one of the formulas we discussed above and place bounds on it:

Since we know that the rod is uniform, we can take the linear mass density out of the integral because it is a constant.

As you can see, the lambdas cancel out! Now we can evaluate the integrals.

Now we can plug in our bounds and simplify. This leads us to:

Hopefully, this answer seems intuitive to you! We'll be seeing problems similar to this when we tackle rotational inertia next unit.

Even though AP Physics 1 is not calculus-based, we can practice applications of the center of mass with FRQs from that test too!

Taken from College Board

**Answer:**

The trick to this question is realizing that it is asking for the * center of mass of the system*. So the speed of it should only change when momentum isn't conserved, meaning when there is impulse!

**Answer:**

Same focus as before, realize it is the center of mass of the system! Think of how you were searching for the x coordinate of the center of mass, you can apply the same strategy for velocity.

ap physics c: mech

🚗 Unit 1: Kinematics

🚀 Unit 2: Newton's Laws of Motion

🎢 Unit 3: Work, Energy, and Power

🎳 Unit 4: Systems of Particles and Linear Momentum

🚲 Unit 5: Rotation

🌊 Unit 6: Oscillations

continue learning

Fiveable Community students are already meeting new friends, starting study groups, and sharing tons of opportunities for other high schoolers. Soon the Fiveable Community will be on a totally new platform where you can share, save, and organize your learning links and lead study groups among other students!🎉

*ap® and advanced placement® are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.

© fiveable 2021 | all rights reserved.

2550 north lake drive

suite 2

milwaukee, wi 53211